Irregular AreasIrregular areas are those areas that do not fall withina definite standard shape. As you already have learned,there are formulas for computing the area of a circle, arectangle, a triangle, and so on. However, we do nothave a standard formula for computing the area of anirregular shaped plane, unless we use highermathematics (calculus), and integrate incremental areasusing lower and upper limits that define theboundaries.As an EA, however, most areas you will beconcerned with are those you will meet in planesurveying. In most surveys, the computed area is thehorizontal projection of the area rather than the actualsurface of the land. The fieldwork in finding areasconsists of a series of angular and linear measurements,defining the outline of whatever the shape is of the areaconcerned, and forming a closed traverse. Thefollowing office computation methods, which you willlearn as you advance in rate, are:1. Plotting the closed traverse to scale andmeasuring the enclosed area directly with a polarplanimeter (used only where approximate results arerequired, or for checking purposes).2. Subdividing the area into a series of triangles,and taking the summation of all the areas of thesetriangles.3. Computing the area using the coordinates of theindividual points of the traverse (called coordinatemethod).4. Computing the area by means of the balancedlatitude and departure, and calculated DOUBLEMERIDIAN DISTANCES of each course (called theDMD method).5. Computing the area by counting squares; thismethod is nothing but just superimposing small squaresplotted on a transparent paper having the same scale asthe plotted traverse (or of known graphical ratio) andcounting the number of squares within the traverse. Thesmaller the squares, the closer to the approximate areayou will get.6. Computing an irregular area bounded by a curveand perpendicular lines, as shown in figure 1-16. Here,you can use the TRAPEZOIDAL RULE. The figure isconsidered as being made up of a series of trapezoids,all of them having the same base and having commonFigure 1-16.-Irregular area by trapezoidal rule.distances between offsets. The formula in computingthe total area is as follows:For the present time, try to find the areas of irregularfigures by subdividing the area to series of triangles andby the method of counting the squares.There are also areas of spherical surfaces and areasof portions of a sphere. For other figures not coveredin this training manual, consult any text on plane andsolid geometry.DETERMINING VOLUMESFrom the preceding section you learned theformulas for computing the areas of various planefigures. These plane areas are important in thecomputation of VOLUMES, as you will see later in thissection.When plane figures are combined to form athree-dimensional object, the resulting figure is1-14